Over 98% of the synapse in the nervous system are formed with neuron dendrites. Postsynaptic currents generated by these synapses must flow through the branches of the dendritic tree to the initial segment, which is generally the locus of synaptic integration. Despite the critical role of passive dendritic propagation in neuronal function and microcircuitry, the propagation losses for synaptic signals is unknown. The degree of correlation between membrane potential in the dendritic terminals with that in the soma or other dendrites is unknown. Under these conditions, investigators cannot interpret the significance of synaptic location, dendritic restructuring in disease states, or the degree to which electrical manipulations in the soma affect the dendritic terminals. All estimates of these parameters are based on mathematical or computational models, but experimentalists need more intricate and flexible analytic models. The purpose of this research project is to advance the usefulness of mathematical models by finding analytic solutions for models that are more realistic. In particular, solutions will be found for neurons with dendrites of arbitrary number, length, diameter, membrane electrical properties. They also are to permit a voltage-clamp or current-clamp to applied at any point on the neuron, and the voltage and current responses to be calculated at any other point, or even the same point. The models are to have at least one dendrite with one or more branch points. Another purpose is to incorporate experimental data into the new models to address the following questions: (1) whether or not somas are and necessary in multipolar models, (2) the magnitude of the electrical length of dendrites, and (3) the magnitude of postsynaptic propagation losses. The long-term goal of this research is to understand the mechanisms of synaptic integration in the moment-by-moment control of motoneurons and muscles, and the role that dendritic structural changes play in the dysfunction and adaptation of nervous tissue to trauma, metabolic diseases, infection, and other neurological conditions.